It's actually both surprisingly meaningful and quite precise in its meaning which also makes it completely unintelligible if you don't know the words it uses.
Ordered field: satisfying the properties of an algebraic field - so a set, an addition and a multiplication with the proper properties for these operations - with a total order, a binary relation with the proper properties.
Usual topology: we will use the most common metric (a function with a set of properties) on R so the absolute value of the difference
Finite-dimentional: can be generated using only a finite number of elements
Commutative: the operation will give the same result for (a x b) and (b x a)
Unital: as an element which acts like 1 and return the same element when applied so (1 x a) = a
R-algebra: a formally defined algebraic object involving a set and three operations following multiple rules
Algebraically closed: a property on the polynomial of this algebra to be respected. They must always have a root. Untrue in R because of negative. That's basically introducing i as a structural necessity.
Admits a notion of differentiation with reasonable spectral behaviour: This is the most fuzzy part. Differentiation means we can build a notion of derivatives for functions on it which is essential for calculus to work. The part about spectral behavior is probably to disqualify weird algebra isomorphic to C but where differentiation behaves differently. It seems redondant to me if you already have a finite-dimentional algebra.
It's not really complicated. It's more about being familiar with what the expression means. It's basically a fancy way to say that if you ask for something looking like R with a calculus acting like the one of functions on R but in higher dimensions, you get C.